R-commutative Geometry and Quantization of Poisson Algebras

An r-commutative algebra is an algebra A equipped with a Yang-Baxter operator R:A ⊗ A → A ⊗ A satisfying m = mR, where m:A ⊗ A → A is the multiplication map, together with the compatibility conditions R(a⊗ 1) = 1 ⊗ a, R(1 ⊗ a) = a ⊗ 1, R(id ⊗m) = (m ⊗ id)R2R1 and R(m ⊗ id) = (id ⊗ m)R1R2. The basic notions of differential geometry extend from commutative (or supercommutative) algebras to r-commutative algebras. Examples of rcommutative algebras obtained by quantization of Poisson algebras include the Weyl algebra, noncommutative tori, quantum groups, and certain quantum vector spaces. In many of these cases the r-commutative de Rham cohomology is stable under quantization.